Lyapunov Exponent of Rank One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution

Abstract

The Lyapunov exponent corresponding to a set of square matrices A = {A 1 , . . . , A n } and a probability distribution p over {1, . . . , n} is λ(A, p) := lim k→∞ 1 k E log A σ k · · · A σ2 A σ1 , where σ i are i.i.d. according to p. This quantity is of fundamental importance to control theory since it determines the asymptotic convergence rate e λ(A,p) of the stochastic linear dynamical system x k+1 = A σ k x k . This paper investigates the following "design problem": given A, compute the distribution p minimizing λ(A, p). Our main result is that it is NP-hard to decide whether there exists a distribution p for which λ(A, p) < 0, i.e. it is NP-hard to decide whether this dynamical system can be stabilized.This hardness result holds even in the "simple" case where A contains only rank-one matrices. Somewhat surprisingly, this is in stark contrast to the Joint Spectral Radius -the deterministic kindred of the Lyapunov exponent -for which the analogous optimization problem for rank-one matrices is known to be exactly computable in polynomial time.To prove this hardness result, we first observe via Birkhoff's Ergodic Theorem that the Lyapunov exponent of rank-one matrices admits a simple formula and in fact is a quadratic form in p. Hardness of the design problem is shown through a reduction from the Independent Set problem. Along the way, simple examples are given illustrating that p → λ(A, p) is neither convex nor concave in general. We conclude with extensions to continuous distributions, exchangeable processes, Markov processes, and stationary ergodic processes.where the σ i are independently and identically distributed (i.i.d.) according to p. Over the past half century, this quantity λ(A, p) has received significant interest due to its many connections and applications to diverse fields including probability, ergodic theory, dynamical systems, functional analysis, representation theory, computer image generation of fractals, control theory, and more; see e.g. the surveys [4,9,10,16,21,43] and references within.The primary application of the Lyapunov exponent that this paper focuses on is in control theory. The fundamental connection is that λ(A, p) dictates the asymptotic growth rate of the stochastic linear dynamical system( 1.2)The authors are with the

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